EXTREMAL FUNCTIONS FOR A SHARP MOSER–TRUDINGER INEQUALITY

2006 ◽  
Vol 17 (03) ◽  
pp. 331-338 ◽  
Author(s):  
YUNYAN YANG
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Smooth Domain ◽  
Extremal Functions ◽  
First Eigenvalue ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Bounded Smooth ◽  
Trudinger Inequality

Let Ω be a bounded smooth domain in ℝ2, and λ1(Ω) the first eigenvalue of the Laplacian with Dirichlet boundary condition in Ω. Then Adimurthi and Druet show that for any 0 ≤ α < λ1(Ω)[Formula: see text] We prove in this paper that there exist extremal functions for the above inequality. In other words, we show that [Formula: see text] is attained for any 0 ≤ α < λ1(Ω).

On the system of p-Laplacian equations with critical growth

2018 ◽  
Vol 29 (02) ◽  
pp. 1850008 ◽  
Author(s):  
Xiangqing Liu ◽  
Junfang Zhao ◽  
Jiaquan Liu
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary ◽  
Critical Growth ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Bounded Smooth ◽  
Sign Changing Solutions ◽  
Laplacian Equations

In this paper, we consider the system of [Formula: see text]-Laplacian equations with critical growth [Formula: see text] where [Formula: see text] is a bounded smooth domain in [Formula: see text] the first eigenvalue of the [Formula: see text]-Laplacian operator [Formula: see text] with the Dirichlet boundary condition, [Formula: see text] for [Formula: see text]. The existence of infinitely many sign-changing solutions is proved by the truncation method and by the concentration analysis on the approximating solutions, provided [Formula: see text].

scholarly journals Isolation and simplicity for the first eigenvalue of thep-Laplacian with a nonlinear boundary condition

2002 ◽  
Vol 7 (5) ◽  
pp. 287-293 ◽  
Author(s):  
Sandra Martínez ◽  
Julio D. Rossi
Keyword(s):  
Boundary Condition ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Smooth Domain ◽  
First Eigenvalue ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Bounded Smooth

We prove the simplicity and isolation of the first eigenvalue for the problemΔpu=|u|p−2uin a bounded smooth domainΩ⊂ℝN, with a nonlinear boundary condition given by|∇u|p−2∂u/∂v=λ|u|p−2uon the boundary of the domain.

scholarly journals On a class of semilinear elliptic equations with boundary conditions and potentials which change sign

2005 ◽  
Vol 2005 (2) ◽  
pp. 95-104
Author(s):  
M. Ouanan ◽  
A. Touzani
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Bounded Smooth

We study the existence of nontrivial solutions for the problemΔu=u, in a bounded smooth domainΩ⊂ℝℕ, with a semilinear boundary condition given by∂u/∂ν=λu−W(x)g(u), on the boundary of the domain, whereWis a potential changing sign,ghas a superlinear growth condition, and the parameterλ∈]0,λ1];λ1is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.

scholarly journals The existence of nontrivial solutions for a critically coupled Schr\”{o}dinger system in a bounded domain of $\R^3$

2021 ◽  
Author(s):  
Hongyu Ye ◽  
Lina Zhang
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
The First Eigenvalue ◽  
Nirenberg Problem

In this paper, we consider the following coupled Schr\”{o}dinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg problem $$\left\{% \begin{array}{ll} -\Delta u+\lambda_1 u=\mu_1 u^5+ \beta u^2v^3, & \hbox{$x\in \Omega$}, \\ -\Delta v+\lambda_2 v=\mu_2 v^5+ \beta v^2u^3, & \hbox{$x\in \Omega$}, \\ u=v=0,& \hbox{$x\in \partial\Omega$}, \\ \end{array}% \right.$$ where $\Omega$ is a ball in $\R^3,$ $-\lambda_1(\Omega)<\lambda_1,\lambda_2<-\frac14\lambda_1(\Omega)$, $\mu_1,\mu_2>0$ and $\beta>0$. Here $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary condition in $\Omega$. We show that the problem has at least one nontrivial solution for all $\beta>0$.

scholarly journals On the extremal solutions of semilinear elliptic problems

2005 ◽  
Vol 2005 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Lamia Ben Chaabane
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Extremal Solutions ◽  
Bounded Smooth Domain ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

We investigate here the properties of extremal solutions for semilinear elliptic equation−Δu=λf(u)posed on a bounded smooth domain ofℝnwith Dirichlet boundary condition and withfexploding at a finite positive valuea.

HARDY-SOBOLEV INEQUALITY IN H1(Ω) AND ITS APPLICATIONS

2002 ◽  
Vol 04 (03) ◽  
pp. 409-434 ◽  
Author(s):  
ADIMURTHI
Keyword(s):  
Boundary Condition ◽  
Sobolev Inequality ◽  
Neumann Boundary Condition ◽  
Remainder Term ◽  
Blow Up ◽  
Neumann Boundary ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Eigen Value ◽  
Bounded Smooth

In this article, we have determined the remainder term for Hardy–Sobolev inequality in H1(Ω) for Ω a bounded smooth domain and studied the existence, non existence and blow up of first eigen value and eigen function for the corresponding Hardy–Sobolev operator with Neumann boundary condition.

scholarly journals Twin Positive Solutions for Schrödinger-Kirchhoff-Type Problem with Singularity and Critical Exponents

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Wei Han ◽  
Yangyang Zhao
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Perturbation Methods ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
The First Eigenvalue ◽  
Kirchhoff Type Problem

We study in this paper the following singular Schrödinger-Kirchhoff-type problem with critical exponent -a+b∫Ω∇u2dxΔu+u=Q(x)u5+μxα-2u+f(x)(λ/uγ) in Ω,u=0 on ∂Ω, where a,b>0 are constants, Ω⊂R3 is a smooth bounded domain, 0<α<1, λ>0 is a real parameter, γ∈(0,1) is a constant, and 0<μ<aμ1 (μ1 is the first eigenvalue of -Δu=μxα-2u, under Dirichlet boundary condition). Under appropriate assumptions on Q and f, we obtain two positive solutions via the variational and perturbation methods.

scholarly journals Weighted Steklov problem under nonresonance conditions

2018 ◽  
Vol 36 (4) ◽  
pp. 87-105
Author(s):  
Jonas Doumatè ◽  
Aboubacar Marcos
Keyword(s):  
Boundary Condition ◽  
Nonlinear Problem ◽  
Weak Solutions ◽  
Existence Results ◽  
Smooth Domain ◽  
Existence Of Weak Solutions ◽  
Steklov Problem ◽  
Bounded Smooth Domain ◽  
Carathéodory Function ◽  
Bounded Smooth

We deal with the existence of weak solutions of the nonlinear problem $-\Delta_{p}u+V|u|^{p-2}u$ in a bounded smooth domain $\Omega\subset \mathbb{R}^{N}$ which is subject to the boundary condition $|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=f(x,u)$. Here $V\in L^{\infty}(\Omega)$ possibly exhibit both signs which leads to an extension  of particular cases in literature and $f$ is a Carathéodory function that satisfies some additional conditions. Finally we prove, under and between nonresonance condtions, existence results for the problem.

scholarly journals Extinction and Nonextinction for the Fast Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chunlai Mu ◽  
Li Yan ◽  
Yi-bin Xiao
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Initial Data ◽  
Diffusion Equation ◽  
Dirichlet Boundary ◽  
Fast Diffusion ◽  
First Eigenvalue ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Fast Diffusion Equation ◽  
The First Eigenvalue

This paper deals with the extinction and nonextinction properties of the fast diffusion equation of homogeneous Dirichlet boundary condition in a bounded domain ofRNwithN>2. For0<m<1, under appropriate hypotheses, we show thatm=pis the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of-Δwith homogeneous Dirichlet boundary.

scholarly journals Blow up of solutions of semilinear heat equations in general domains

2015 ◽  
Vol 17 (02) ◽  
pp. 1350042 ◽  
Author(s):  
Valeria Marino ◽  
Filomena Pacella ◽  
Berardino Sciunzi
Keyword(s):  
Boundary Condition ◽  
Initial Data ◽  
Heat Equation ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Initial Value ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

Consider the nonlinear heat equation vt - Δv = |v|p-1v in a bounded smooth domain Ω ⊂ ℝn with n > 2 and Dirichlet boundary condition. Given up a sign-changing stationary classical solution fulfilling suitable assumptions, we prove that the solution with initial value ϑup blows up in finite time if |ϑ - 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent [Formula: see text]. Since for ϑ = 1 the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped with respect to the origin. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.

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